Several versions of the compensated compactness principle

S. E. Pastukhova   ; A. S. Khripunova

Sbornik. Mathematics, Tome 202 (2011) no. 9, pp. 1387-1412

Résumé

The convergence of the product of a solenoidal vector $w_\varepsilon$ and a gradient $\nabla u_\varepsilon$ in $L^1(\Omega)$ (where $\Omega$ is a region in $\mathbb R^d$) is investigated in the case when the factors converge weakly in the spaces $L^\gamma(\Omega)^d$ and $L^\alpha(\Omega)^d$, respectively, with $1/\gamma+1/\alpha>1$, which means that the main assumption of the classical $div$-$curl$ lemma fails. Nevertheless, the same convergence (in the sense of distributions in $\Omega$) $$ \lim_{\varepsilon\to0}w_\varepsilon\cdot\nabla u_\varepsilon =\lim_{\varepsilon\to0}w_\varepsilon\cdot\lim_{\varepsilon\to0} \nabla u_\varepsilon=w\cdot\nabla u $$ as in the framework of the $div$-$curl$ lemma, survives under certain additional assumptions. The new versions of the compensated compactness principle proved in the paper can be used in homogenization and in the theory of $G$-convergence of monotone operators with non-standard coercivity and growth properties, for instance, some degenerate operators. Bibliography: 20 titles.

Bibliographie

[1] F. Murat, “Compacité par compensation”, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 5:3 (1978), 489–507 | MR | Zbl

[2] V. V. Zhikov, S. E. Pastukhova, “On the compensated compactness principle”, Dokl. Math., 82:1 (2010), 590–595 | DOI | MR | Zbl

[3] V. V. Zhikov, “On the technique for passing to the limit in nonlinear elliptic equations”, Funct. Anal. Appl., 43:2 (2009), 96–112 | DOI | MR

[4] V. V. Zhikov, S. E. Pastukhova, “Improved integrability of the gradients of solutions of elliptic equations with variable nonlinearity exponent”, Sb. Math., 199:12 (2008), 1751–1782 | DOI | MR | Zbl

[5] V. V. Zhikov, “On the weak convergence of fluxes to a flux”, Sb. Math., 81:1 (2010), 58–62 | DOI | MR | Zbl

[6] V. V. Zhikov, S. E. Pastukhova, “Lemmas on compensated compactness in elliptic and parabolic equations”, Proc. Steklov Inst. Math., 270 (2010), 104–131 | DOI | MR | Zbl

[7] V. V. Zhikov, S. M. Kozlov, O. A. Oleinik, Homogenization of differential operators and integral functionals, Springer-Verlag, Berlin, 1994 | MR | MR | Zbl | Zbl

[8] V. V. Zhikov, “On passage to the limit in nonlinear variational problems”, Russian Acad. Sci. Sb. Math., 76:2 (1993), 427–459 | DOI | MR | Zbl | Zbl

[9] V. V. Zhikov, “Lavrent'ev effect and the averaging of nonlinear variational problems”, Differential Equations, 27:1 (1991), 32–39 | MR | Zbl

[10] I. Ekeland, R. Temam, Convex analysis and variational problems, Studies in Mathematics and its Applications, 1, North-Holland, Amsterdam–Oxford; Elsevier, New York, 1976 | MR | MR | Zbl | Zbl

[11] V. V. Zhikov, “On Lavrent'ev's effect”, Dokl. Math., 52:3 (1995), 325–329 | MR | Zbl

[12] V. V. Zhikov, “On Lavrentiev's phenomenon”, Russian J. Math. Phys., 3:2 (1995), 249–269 | MR | Zbl

[13] A. Pankov, $G$-convergence and homogenization of nonlinear partial differential operators, Math. Appl., 422, Kluwer Acad. Publ., Dordrecht–Boston–London, 1997 | MR | Zbl

[14] V. V. Zhikov, S. E. Pastukhova, “Homogenization of degenerate elliptic equations”, Siberian Math. J., 49:1 (2008), 80–101 | DOI | MR | Zbl

[15] R. R. Coifman, C. Fefferman, “Weighted norm inequalities for maximal functions and singular integrals”, Studia Math., 51 (1974), 241–250 | MR | Zbl

[16] E. B. Fabes, C. E. Kenig, R. P. Serapioni, “The local regularity of solutions of degenerate elliptic equations”, Comm. Partial Differential Equations, 7:1 (1982), 77–116 | DOI | MR | Zbl

[17] V. V. Zhikov, “Weighted Sobolev spaces”, Sb. Math., 189:8 (1998), 1139–1170 | DOI | MR | Zbl

[18] R. de Archangelis, F. Serra Cassano, “On the homogenization of degenerate elliptic equations in divergence form”, J. Math. Pures Appl. (9), 71:2 (1992), 119–138 | MR | Zbl

[19] T. Gallouët, J. Lederer, R. Lewandowski, F. Murat, L. Tartar, “On a turbulent system with unbounded eddy viscosities”, Nonlinear Anal., 52:4 (2003), 1051–1068 | DOI | MR | Zbl

[20] V. I. Bogachev, Differentiable measures and the Malliavin calculus, Math. Surveys Monogr., 164, Amer. Math. Soc., Providence, RI, 2010 | MR | Zbl